**2.5 Conservation of Momentum: The Navier-Stokes Equations of Motion**

**2.5.1 Cartesian Coordinates**

Application of Newton’s law of motion to the

element shown in Fig. 2.5, gives

Application of (a) in the *x*-direction, gives

Each term in (b) is expressed in terms of flow

field variables: density, pressure, and velocity components:

**Mass of the element:**

**IMPORTANT**

**THE NORMAL AND SHEARING STRESSES ARE EXPRESSED IN TERMS**

**OF VELOSICTY AND PRESSURE. THIS IS VALID FOR NEWTOINAN**

**FLUIDS. (See equations 2.7a-2.7f).**

**THE RESULTING EQUATIONS ARE KNOWN AS THE NAVIER-STOKES**

**EQUAITONS OF MOTION**

**SPECIAL SIMPLIFIED CASES:**

**2.5.2 Cylindrical Coordinates**

The three equations corresponding to (2.10) in cylindrical coordinates are (2.11*r*),

and (2.11*z*).

**2.5.3 Spherical Coordinates**

The three equations corresponding to (2.10) in spherical coordinates are (2.11*r*), and